null hypothesis testing

Imagine an interferometer as a giant ear listening to the radio whisper of a distant galaxy. Three antennas spread across continents capture the signal, and the difference in their phases combines into a quantity that is immune to atmospheric jitter. This quantity is called the closure phase.

Two tiny diamonds float in a vacuum, separated by a distance that light cannot cross during the experiment. Inside each, like a frozen spark, resides an entangled spin — a quantum state linking them into a single whole. Now the diamonds begin to move, but not just any motion: each one seems to split, existing in two places at once.

In a warehouse, there are boxes of apples and pears. There are many apples and few pears. But when the stock clerk says "a fruit with smooth skin," we are somehow more likely to recall the rare pear than the common apple. This is the inverse base-rate effect — a persistent glitch in how the brain resolves ambiguity between categories.

A glucose meter that doesn't prick your finger but just looks at your wrist: that's the dream for millions of people with diabetes. It's the promise of algorithms that extract glucose levels from the ordinary pulse signal, the photoplethysmogram (the same one that measures heart rate in smartwatches).

We're used to thinking that noticing only confirmations of our guesses is a mistake. But recent calculations, inspired by wave behavior, flip that view. It turns out that the habit of seeking the familiar is the best way to reduce information noise (scientists call it entropy).

In 1953, mathematician Alexander Grothendieck proved a theorem that became a cornerstone in the theory of tensor products. It states that for ordinary function spaces, two natural norms, the projective and the injective, are equivalent up to a universal constant. In other words, two ways of measuring the 'size' of combinations of vectors give similar results.

The Universe is expanding faster and faster thanks to mysterious dark energy, discovered in the late 1990s by Adam Riess, Saul Perlmutter, and Brian Schmidt using supernovae.

Two particles behave like a pair of unusual dice: thrown at opposite ends of the universe, they always land on the same face. Physicist John Stewart Bell calculated: if the dice had hidden rules, the frequency of matches would have a strict limit.

The weak equivalence principle, silently tested since the time of Galileo and laid at the foundation of general relativity, states: all bodies fall equally, regardless of their internal composition.

Twenty-first-century astrophysics is deafening with triumphs: we’ve caught the tremor of spacetime from the merger of invisible giants; glimpsed the 'shadow' of a supermassive monster in a distant galaxy; traced the orbits of stars dancing around a compact something at the heart of the Milky Way.

The hunt for life on distant planets is like diagnosing a rare disease with a test of unknown accuracy. A single positive result means nothing: maybe it's a glitch? We search the atmospheres of exoplanets (worlds around other stars) for traces of water and other compounds, but any signal could be a false alarm.

Light bends around massive bodies: this has been known since Einstein's time. But what if gravity itself is not a smooth fabric of space but a stream of tiny carrier particles? Then a beam passing near a particle should not just bend but enter into a special quantum relationship with it. This relationship is called entanglement.

A quantum system can possess a property called magic. It's not about tricks: it's a measure of how hard it is to reproduce its behavior on a classical computer. Ordinary states, known as stabilizer states, are easy for a computer to handle. But states with magic leave it stumped.

An epidemic is not just a virus: it's millions of personal decisions. One person keeps riding the subway; another locks themselves at home. Classical models describe the average behavior of a crowd but don't ask why each individual acts that way.

Comparing two scents is easy even without knowing their ingredients, just tell if they're alike. This everyday principle became the foundation for a new class of Bell inequalities, reimagining tests of the spooky connection between particles. Previously, you had to measure precise properties, like spin direction, but the authors showed that simply comparing outcomes is enough.

In a dark chamber where the temperature sits just four degrees above absolute zero, a sapphire crystal the size of a grain of rice lies waiting. When a neutrino, a ghostly particle that barely notices matter, flies through it, the crystal shudders and warms by a tiny fraction of a degree.

A coin toss seems random, but in reality, physics decides everything: the force of the throw, air resistance, the surface's elasticity. If you knew all the details, you could predict the outcome. True randomness lives deeper — in the quantum world.

The universe is expanding, but at what rate is a question that has been troubling astronomers for decades. Two methods of measurement give different answers, and this discrepancy is called the Hubble tension. Some look at the early universe through the cosmic microwave background, others at nearby supernovae and Cepheid variables.

Genetic diagnostics is like comparing two photos taken with different cameras. If one camera slightly shifts the colors, the image will show "defects" that don't exist in reality.

All bodies fall the same: this was first observed by Galileo. Einstein elevated the observation to the equivalence principle: gravity and acceleration are indistinguishable, giving birth to curved spacetime.

In 1935, Einstein, Podolsky, and Rosen proposed a thought experiment that, in their view, exposed the incompleteness of quantum mechanics. Two particles fly apart after interacting, and measuring the momentum of one instantly gives exact knowledge of the other's momentum.

An orchestra without a conductor plays in harmony, and you want to understand who influences whom. The violinist hears the cello, the cellist catches the rhythm of the percussion, and the flute adapts to everyone at once.